Line L1 has equation 14x + 4y = 5. Line L2 is parallel to L1 and intersects the x-axis at...

1. TRANSLATE the problem information

• Given information:
  • L1 has equation \(14\mathrm{x} + 4\mathrm{y} = 5\)
  • L2 is parallel to L1
  • L2 has x-intercept \((\mathrm{a}, 0)\) and y-intercept \((0, \mathrm{b})\)
  • Need to find \(\frac{\mathrm{b}}{\mathrm{a}}\)

2. INFER the key relationship

• Since L2 is parallel to L1, they must have the same slope
• To find this common slope, I need to determine L1's slope first
• Strategy: Convert L1 to slope-intercept form, then use L2's intercepts to express its slope

3. SIMPLIFY L1's equation to find its slope

• Starting with: \(14\mathrm{x} + 4\mathrm{y} = 5\)
• Isolate y: \(4\mathrm{y} = -14\mathrm{x} + 5\)
• Divide by 4: \(\mathrm{y} = \frac{-14}{4}\mathrm{x} + \frac{5}{4}\)
• SIMPLIFY the fraction: \(\mathrm{y} = \frac{-7}{2}\mathrm{x} + \frac{5}{4}\)
• Therefore, slope of L1 = \(\frac{-7}{2}\)

4. INFER L2's slope using its intercept points

• L2 passes through \((\mathrm{a}, 0)\) and \((0, \mathrm{b})\)
• Using slope formula: \(\mathrm{slope} = \frac{\mathrm{y}_2 - \mathrm{y}_1}{\mathrm{x}_2 - \mathrm{x}_1}\)
• slope of L2 = \(\frac{\mathrm{b} - 0}{0 - \mathrm{a}} = \frac{\mathrm{b}}{-\mathrm{a}} = \frac{-\mathrm{b}}{\mathrm{a}}\)

5. SIMPLIFY to find b/a

• Since slopes are equal: \(\frac{-\mathrm{b}}{\mathrm{a}} = \frac{-7}{2}\)
• Multiply both sides by -1: \(\frac{\mathrm{b}}{\mathrm{a}} = \frac{7}{2}\)

Answer: D (7/2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students forget that parallel lines have equal slopes, or don't make the connection to use the intercept points to find L2's slope. They might try to find L2's equation directly without using the parallel line relationship, leading to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students set up the slope formula incorrectly with the intercept points. They might calculate slope as \(\frac{0 - \mathrm{b}}{\mathrm{a} - 0} = \frac{-\mathrm{b}}{\mathrm{a}}\) instead of \(\frac{\mathrm{b} - 0}{0 - \mathrm{a}} = \frac{-\mathrm{b}}{\mathrm{a}}\), getting a positive result. This may lead them to select Choice (D) 7/2 but for the wrong reason, or Choice (A) -7/2 if they don't resolve the sign error.

The Bottom Line:

This problem requires connecting the abstract concept of parallel lines to concrete intercept points. Students who can't bridge this gap often struggle to even begin a systematic solution.